David J. Green

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Cohomology of group number 548 of order 128

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Ring generators

Ring relations

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General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  − 1

  (t  −  1)3

• The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 minimal generators of maximal degree 4:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_1_2, an element of degree 1
4. b_2_3, an element of degree 2
5. b_2_4, an element of degree 2
6. c_2_5, a Duflot regular element of degree 2
7. b_3_9, an element of degree 3
8. c_4_14, a Duflot regular element of degree 4

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Ring relations

There are 10 minimal relations of maximal degree 6:

1. a_1_0²
2. a_1_1² + a_1_0·a_1_1
3. a_1_0·b_1_2
4. b_2_3·a_1_0
5. b_2_4·b_1_2 + b_2_3·b_1_2 + b_2_4·a_1_0
6. b_2_3·b_1_2² + b_2_3² + c_2_5·b_1_2²
7. b_1_2·b_3_9 + b_2_3·b_1_2² + b_2_3·b_2_4 + c_2_5·b_1_2²
8. b_2_3·b_1_2² + b_2_3·b_2_4 + a_1_0·b_3_9 + c_2_5·b_1_2²
9. b_2_3·b_3_9 + b_2_4²·a_1_0
10. b_3_9² + b_2_4³ + b_2_3³

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 6.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_2_5, a Duflot regular element of degree 2
  2. c_4_14, a Duflot regular element of degree 4
  3. b_1_2² + b_2_4 + b_2_3, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. c_2_5 → c_1_1², an element of degree 2
7. b_3_9 → 0, an element of degree 3
8. c_4_14 → c_1_1⁴ + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. b_2_3 → c_1_1·c_1_2, an element of degree 2
5. b_2_4 → c_1_1·c_1_2, an element of degree 2
6. c_2_5 → c_1_1·c_1_2 + c_1_1², an element of degree 2
7. b_3_9 → 0, an element of degree 3
8. c_4_14 → c_1_1²·c_1_2² + c_1_1³·c_1_2 + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → c_1_2², an element of degree 2
6. c_2_5 → c_1_2² + c_1_1², an element of degree 2
7. b_3_9 → c_1_2³, an element of degree 3
8. c_4_14 → c_1_1²·c_1_2² + c_1_1⁴ + c_1_0²·c_1_2² + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128